holdsForLocalizationAway — Mathlib

English

The composition of smooth ring homomorphisms is smooth.

Русский

Сложение гладких гомоморфизмов сохраняет гладкость.

LaTeX

$$$\text{comp}\;\{f:\,R\to S\}\;\{g:\,S\to T\} \; (hf:\,f.Smooth) \; (hg:\,g.Smooth) \Rightarrow (g.f).Smooth$$

Lean4

theorem holdsForLocalizationAway : HoldsForLocalizationAway Smooth :=
  by
  introv R h
  rw [smooth_algebraMap]
  exact ⟨Algebra.FormallySmooth.of_isLocalization (.powers r), IsLocalization.Away.finitePresentation r⟩

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